Form Factors in D Meson Decays Dae Sung Hwang(a) and Do-Won Kim(b) a: Department of Physics, Sejong University, Seoul 143–747, Korea b: Department of Physics, Kangnung National University, Kangnung 210-702, Korea 9 9 9 1 n a Abstract J 5 We study the dΓ/dq2 spectra and the branching fractions of the D meson exclusive 2 v semileptonic decays with the lepton mass effects into consideration. We investigate 2 6 3 their sensitivity to form factor models, and find that the decays to a pseudoscalar 6 0 meson and a lepton pair are sensitive to the property of the formfactor F (q2), and 1 8 9 those to a vector meson and a lepton pair to the form factor A (q2). We also an- / 1 h p alyze the experimental results of the branching fractions (D0 K−(or π−) π+) - B → p and (D0 K−(or π−) e+ν), and show that it is implied that F (q2) is of dipole e B → 1 h type, instead of simple pole type which is commonly assumed. : v i X r a PACS codes: 13.20.Fc, 13.20.-v, 13.25.Ft, 14.40.Lb Key words: D Meson, Semileptonic Decay, Lepton Mass Effect, Form Factor ae-mail: [email protected] be-mail: [email protected] 1. Introduction The CP-violation phenomenon is discovered only in the K ππ decay and L → the charge asymmetry in the decay K π±l∓ν for more than 30 years. The L → B-factories at KEK and SLAC are under construction for the discovery of CP- violation in the B meson system. The mechanism of CP-violation through the complex phase of the CKM three family mixing matrix [1] is presently considered standard for the CP-violation. In order to measure the CKM matrix elements accurately, it is important to know the hadronic form factors of the transition ma- trix elements reliably. For the heavy to heavy transitions the heavy quark effective theory provides good informations for the form factors. However, for the heavy to light transitions the understanding of the form factors is still limited and this fact hinders the extractions of the CKM matrix elements fromexperimental results significantly. At the same time, we will be able to have important clues for the internal structures of hadrons by knowing these form factors well. Semileptonic decay processes are good sources for the knowledge of the form factors both exper- imentally and theoretically, and the lepton mass effects in heavy meson exclusive semileptonic decays were studied by K¨orner and Schuler [2]. We derive the formulas for dΓ/dq2 with non-zero lepton mass in the forms which are efficient to study the form factor dependences. This formula for the pseudoscalar to pseudoscalar transition was also given by Khodjamirian et al. [3]. By using these formulas we study the dΓ/dq2 spectra and branching fractions of the exclusive semileptonic D meson decays: D0 K(∗)−e+ν, D0 K(∗)−µ+ν, → → D0 π−(or ρ−) e+ν and D0 π−(or ρ−) µ+ν. In this anslysis we employ three → → models of form factors and show how the results are influenced by the difference of form factors. For the decays to a pseudoscalar meson and a lepton pair, the results are sensitive to the property of the form factor F (q2). For the decays 1 to a vector meson and a lepton pair, the results are sensitive to the form factor 2 A (q2), and not to the other ones (A (q2), A (q2) and V(q2)). We also analyze 1 0 2 the experimental results of the branching fractions (D0 K−(or π−) π+) and B → (D0 K−(or π−) e+ν), and show that it is implied that the form factor F (q2) 1 B → of the D to K and the D to π transitions are of dipole type, instead of simple pole type which is commonly assumed in the studies of the D meson decays. 2. Semileptonic Decays of Heavy Mesons From Lorentz invariance one finds the decomposition of the hadronic matrix element for pseudoscalar to pseudoscalar meson transition in terms of hadronic form factors: < P(p) J P(P) > µ | | = (P +p) f (q2)+(P p) f (q2) µ + µ − − M2 m2 M2 m2 = (P +p) − q F (q2)+ − q F (q2), (1) µ − q2 µ 1 q2 µ 0 (cid:16) (cid:17) where J = q¯′γ (1 γ )q. We use the following notations: M represents initial µ µ 5 − meson mass, m final meson mass, m lepton mass, P initial meson momentum, p l final meson momentum, and q = (P p) . The form factors F (q2) and F (q2) µ µ 1 0 − correspond to 1− and 0+ exchanges, respectively. At q2 = 0 we have the constraint F (0) = F (0), since the hadronic matrix element in (1) is nonsingular at this 1 0 kinematic point. The q2 distribution of the semileptonic decay D0 K−l+ν is given in terms → of the hadronic form factors F (q2) and F (q2) as: 1 0 dΓ(D0 K−l+ν) G2 m2 → = F V 2K(q2)(1 l )2 (2) cs dq2 24π3 | | − q2 × 1m2 m2 3 m2 [(K(q2))2(1+ l ) F (q2) 2 + M2(1 )2 l F (q2) 2], 2 q2 | 1 | − M2 8 q2 | 0 | where K(q2), momentum of the final meson in the D meson rest frame, is given by 1 1 K(q2) = (M2 +m2 q2)2 4M2m2 2, (3) 2M − − (cid:16) (cid:17) 3 and the physically allowed range of q2 is given by m2 q2 (M m)2. (4) l ≤ ≤ − For m = 0, (2) is reduced to the commonly used well-known formula: l dΓ(D0 K−l+ν) G2 → = F V 2(K(q2))3 F (q2) 2, (5) dq2 24π3 | cs| | 1 | and 0 q2 (M m)2. We note in (5) that only F (q2) contributes for m = 0, 1 l ≤ ≤ − however, for m = 0 F (q2) also contributes as we can see in (2). l 0 6 From Lorentz invariance one finds the decomposition of the hadronic matrix element for pseudoscalar to vector meson transition in terms of hadronic form factors: < V(p) J P(P) > µ | | P P q P = ε∗ν(p) (M +m)g A (q2) 2 µ ν A (q2)+ µ ν A (q2) µν 1 2 3 − M +m M +m (cid:16) Pρpσ +iε V(q2) , (6) µνρσ M +m (cid:17) where ε = 1 and 0123 M2 m2 +q2 q2 2mA (q2) = (M +m)A (q2) − A (q2)+ A (q2). (7) 0 1 2 3 − M +m M +m The form factors V(q2), A (q2), A (q2) and A (q2) correspond to 1−, 1+, 1+ and 1 2 0 0− exchanges, respectively. At q2 = 0 we have the constraint 2mA (0) = (M + 0 m)A (0) (M m)A (0), since the hadronic matrix element in (16) is nonsingular 1 2 − − at this kinematic point. After a rather lengthy calculation, the q2 distribution of the semileptonic decay D0 K∗−l+ν is given in terms of the hadronic form factors A (q2), A (q2), A (q2) 1 2 3 → and V(q2) as [4]: dΓ(D0 K∗−l+ν) G2 1 m2 → = F V 2 K(q2)(1 l )2 (8) cs dq2 32π3 | | M2 − q2 × (M +m)2 1 m2 m2 1 A (q2) 2 [ (MK)2(1 l )+q2m2 +(MK)2 l + m2m2] {| 1 | m2 3 − q2 q2 2 l 4 M2 m2 q2 2 m2 m2 +Re(A (q2)A∗(q2))[ − − [ (MK)2(1 l )+2(MK)2 l 1 2 − m2 3 − q2 q2 1 + (M2 +m2 q2)m2]+(M2 m2 +q2)m2] 2 − l − l 1 4 m2 m2 + A (q2) 2 (MK)2[ (MK)2(1 l )+4(MK)2 l +2M2m2] | 2 | (M +m)2m2 3 − q2 q2 l q2 8 m2 m2 + V(q2) 2 [ (MK)2(1 l )+4(MK)2 l ] | | (M +m)2 3 − q2 q2 q2 1 + A (q2) 2 (MK)2m2 | 3 | (M +m)2m22 l 1 Re(A (q2)A∗(q2)) (M2 m2 +q2)(MK)2m2 − 3 2 (M +m)2m2 − l 1 +Re(A (q2)A∗(q2)) (MK)2m2 . 3 1 m2 l} When we take m 0 in (8), it agrees with the formula for m = 0 given in Refs. l l → [5, 6]: dΓ(D0 K∗−l+ν) G2 q2 → = F V 2 K(q2)( H+(q2) 2 + H−(q2) 2 + H0(q2) 2), cs dq2 96π3 | | M2 | | | | | | (9) where 1 4M2K2 H0(q2) = − (M2 m2 q2)(M +m)A (q2) A (q2) , (10) 1 2 2m√q2 − − − M +m (cid:16) (cid:17) 2MK H±(q2) = (M +m)A (q2) V(q2) . 1 − ∓ M +m (cid:16) (cid:17) In the case of the B to D meson (heavy to heavy) transition, the heavy quark effectivetheory(HQET)givestheusefulrelationsbetweentherelevantformfactors [7]: M +m F (q2) = V(q2) = A (q2) = A (q2) = (y), (11) 1 0 2 2√Mm F 2√Mm y +1 F (q2) = A (q2) = (y), 0 1 M +m 2 F where y = (M2 +m2 q2)/(2Mm) = E /m (E is the energy of D(∗) meson D(∗) D(∗) − in the B meson rest frame), and (y) is a form factor which becomes the Isgur- F Wise function in the infinite heavy quark mass limit. When we use the relations 5 (11), for m = 0 the formula (8) becomes the well-known formula for the B to D∗ l transition: dΓ(B¯0 D∗+l−ν¯) G2 → = F V 2m3(M m)2 y2 1(y +1)2 cb dq2 48π3 | | − − × q 4y 1 2yr+r2 1+ − ( D∗(y))2, (12) { y +1 (1 r)2 } F − where r = m/M. 3. D0 K( ) l+ν ∗ − → For the form factors concerned with the exclusive semileptonic decays of D meson, we can not use the relations (11) of the HQET. Therefore, in the study of D meson decays we use models for form factors. The pole-dominance idea suggests the following q2 dependence of the form factors [8]: 1 f (q2) = f (0) , (13) i i (1− mq22fi)nfi wheren andm arecorrespondingpowerandpolemassoftheformfactorsf (q2), fi fi i respectively. The WSB model [8] adopts n = 1. However, the exact values of n fi fi are not known. The relations (11) of the HQET gives the following approximate relation among the powers of the form factors for the heavy to heavy transitions: n = n = n = n = n +1 = n +1. (14) F1 V A0 A2 F0 A1 Non-perturbative analysis of QCD [9] suggests the same relation as (14) for the form factors of the heavy to light transitions. The lattice calculations also show that the form factors F , V and A are more rapidly increasing functions of q2 1 0 than the form factors F and A [6, 10], which favors the relation (14). Therefore, 0 1 we will adopt two other models incorporating the relation (14), as well as the WBS model which assumes n = 1 [8], for the study of the exclusive semileptonic fi decays of D meson. We organize in Table 1 the values of the powers n of the fi 6 three models which we use in this work. For the values of the pole masses and those of the form factors at q2 = 0, we use the values organized in Table 2 and 3, which were given by Wirbel, Stech and Bauer [8]. Their precise values are not known and they should be different for each of the three models. However, their exact values are not crucial for our work of clarifying the model dependences of the dΓ/dq2 spectra and the branching fractions. For D0 K−l+ν, we use the formula (2) with non-zero lepton mass, instead of → the commonly used formula (5) which is true for zero lepton mass. The obtained dΓ(D0 K−l+ν)/dq2 spectrum and branching fractions are presented in Figure → 1 and Table 4, for each of the three models we adopt in this work: WSB, Model I and Model II explained in Table 1. We find that the shape of spectrum of Model II is different from those of WSB and Model I in Figure 1. That is, the value of n determines the shape of spectrum. The experimental result of the E687 F1 CollaborationforthisspectrumwasgiveninFig. 3(a)ofRef. [13],andtheshapeof their spectrum favors Model II better than WSB and Model I. In the experimental extraction of the value of FDK(0), the simple pole of the form factors has been 1 commonly assumed [11, 12]. Under this assumption, FDK(0) = 0.75 0.02 0.02 1 ± ± was extracted [11] from the experimentally measured branching fraction (D0 B → K−e+ν) = (3.68 0.21) 10−2. (InRef. [12], FDK(0) = 0.76 0.03was presented.) ± × 1 ± However, if we assume in this extraction Model II which has the dipole form factor for F (q2), we would get 1 3.49 10−2 FDK(0) = 0.75 × = 0.75 0.85 = 0.64 (15) 1 ×s4.78 10−2 × × for the mean value. In (15) we used our results in Table 4 of the branching fraction (D0 K−e+ν) for WSB (n = 1) and Model II (n = 2), which were obtained B → F1 F1 by using the same value of FDK(0). That is, the experimentally extracted value 1 of FDK(0) is much dependent on the form factor models used in the analysis. In 1 reality, it is not yet established which type of form factors is the right one. For D0 K∗−l+ν, we use the formula (8) with non-zero lepton mass. The → 7 obtained spectra and branching fractions are presented in Figure 2 and Table 5. We find that the results of WSB and Model II are almost the same, and they are significantly different from the results of Model I. This fact implies that the value of n mainly determines the spectra and branching fractions of D0 K∗−l+ν. A1 → 4. D0 π (or ρ ) l+ν − − → For D0 π−l+ν, we use the formula (2) with the replacement of V by V . cs cd → The obtained spectra and branching fractions are presented in Fig. 3 and Table 6. We find that the branching fractions of Model II in Fig. 3 are about twice those of WSB and Model I. Therefore, in case that we determine the value of FDπ(0) from an experimentally measured branching fraction of D0 π−l+ν, the 1 → value of FDπ(0) determined with Model II will be about 1/√2 times its value 1 determined with WSB or Model I. From Table 4 and 6, we also find that the ratio (D0 π−l+ν)/ (D0 K−l+ν) from Model II is pretty bigger than that from B → B → WSB or Model I, but its present experimental result 0.101 0.020 0.003 [14] can ± ± not discriminate them yet. For D0 ρ−l+ν, we use the formula (8) with the replacement of V by V . cs cd → The obtained spectra and branching fractions are presented in Fig. 4 and Table 7. We find in Fig. 4 and Table 7 that the results of WSB and Model II are almost the same, and they are much different from the results of Model I. Therefore, like the D0 K∗−l+ν case, the property of the form factor A (q2) mainly determines 1 → the spectra and branching fractions, and the results are not sensitive to the other form factors (A (q2), A (q2) and V(q2)). 0 2 5. Implications of Experimental Results In this section we compare the exclusive semileptonic decays and the two-body 8 hadronic decays. We start by recalling the relevant effective weak Hamiltonian for the two-body hadronic decay D0 K−π+: → G = FV∗V [C (µ) +C (µ) ] + H.C., (16) Heff √2 cs ud 1 O1 2 O2 = (u¯Γρd)(s¯Γ c), = (s¯Γρd)(u¯Γ c), (17) 1 ρ 2 ρ O O where G is the Fermi coupling constant, V and V are corresponding Cabibbo- F cs ud Kobayashi-Maskawa (CKM) matrix elements and Γ = γ (1 γ ). The Wilson ρ ρ 5 − coefficients C (µ) and C (µ) incorporate the short-distance effects arising from 1 2 the renormalization of from µ = m to µ = O(m ). By using the Fierz eff W c H transformation under which V A currents remain V A currents, we get the − − following equivalent forms: 1 C +C = (C + C ) +C (s¯ΓρTad)(u¯Γ Tac) 1 1 2 2 1 2 1 2 ρ O O N O c 1 = (C + C ) +C (u¯ΓρTad)(s¯Γ Tac), (18) 2 1 2 1 ρ N O c where N = 3 is the number of colors and Ta’s are SU(3) color generators. The c second terms in (18) involve color-octet currents. In the factorization assumption, these terms are neglected and is rewritten in terms of “factorized hadron eff H operators” [8]: G = FV∗V a [u¯Γρd] [s¯Γ c] +a [s¯Γρd] [u¯Γ c] + H.C., (19) Heff √2 cs ud 1 H ρ H 2 H ρ H (cid:16) (cid:17) where the subscript H stands for hadronic implying that the Dirac bilinears inside the brackets be treated as interpolating fields for the mesons and no further Fierz- reordering need be done. The phenomenological parameters a and a are related 1 2 to C and C by a = C + 1 C and a = C + 1 C . The numerical values of a 1 2 1 1 Nc 2 2 2 Nc 1 1 and a for D meson decays are given by [15] 2 a = 1.10 0.05, a = 0.49 0.04. (20) 1 2 ± − ± For the two body decay, in the rest frame of initial meson the differential decay rate is given by 1 p dΓ = 2| 1|dΩ, (21) 32π2|M| M2 9 [(M2 (m +m )2)(M2 (m m )2)]1 p = − 1 2 − 1 − 2 2, (22) 1 | | 2M where M is the initial meson mass, m and m the final meson masses, and p the 1 2 1 momentum of one of the final mesons in the initial meson rest frame. By using (1), (19) and < 0 Γµ π−(q) >= iqµfπ−, (21) gives the following formula for the | | branching ratio of the process D0 K−π+: → G m2 1 m f2 (D0 K−π+) = ( F D)2 V 2 D a2 π V FDK(m2) 2 B → √2 | ud| 8π Γ 1 m2 | cs 0 π | D D 1 m2K 2 1 [ 1 (mK +mπ)2 1 (mK −mπ)2 ]21 . (23) × − m2 2 − m − m D D D (cid:16) (cid:17) (cid:16) (cid:17)(cid:16) (cid:17) On the other hand, from (5) and (13) the branching ratio (B0 K−e+ν) is B → given by G m2 m 2 (D0 K−e+ν) = ( F D)2 D V FDK(0) 2 IDK, (24) B → √2 Γ 192π3 | cs 1 | × D where the dimensionless integral IDK is given by IDK = (1−mmKD)2 dx (1+ mm2K2D −x)2 −4mm2K2D 32. (25) Z0 (cid:16) 1 m2D x 2nF1 (cid:17) − m2F1 (cid:16) (cid:17) In the above, we neglected the electron mass. From (23) and (25) we have (D0 K−π+) f2 m2 2 B → = 6π2 V 2 π 1 K (D0 K−e+ν) | ud| m2 − m2 B → D (cid:16) D(cid:17) m +m m m V FDK(m2) 2 a2 [ 1 ( K π)2 1 ( K − π)2 ]21 | cs 0 π | 1 × − m − m V FDK(0) 2 IDK (cid:16) D (cid:17)(cid:16) D (cid:17) | cs 0 | a2 = 0.225 1 , (26) × IDK where we used the fact FDK(m2) FDK(0) and the following experimetal values 0 π ≃ 0 [16]: mD = mD0 = 1.8646 0.0005 GeV, mK = mK− = 493.677 0.013 MeV, ± ± m = m = 139.56995 0.00035 MeV, f = f = 131.74 0.15 MeV and π π+ π π+ ± ± V = 0.9753 0.0008. ud ± 10